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The Drude model describes the dependence of σ on the lifetime (τ) between scattering events, the free carrier concentration (n), the carrier effective mass (m), and the elementary charge (q): σ=nq2τ/m. For metals the electron–phonon scattering rate increases with T, producing a linear dependence σ−1τ−1T at sufficiently high T. Elementary quantum theory dictates that this relationship cannot continue indefinitely. According to Heisenberg’s uncertainty principle the uncertainty ΔE of a particle’s energy is inversely proportional to its lifetime: ΔE =h/τ. Therefore there exists a minimum τ below which the concept of a well-defined quasiparticle energy becomes unphysical. This lower bound on τ implies a minimum metallic conductivity (σMIR), which is called the Mott–Ioffe–Regel (MIR) limit1,2. Most metals reach their melting temperature well before the MIR limit. There are some so-called ‘saturating’ metals for which σ (T) approaches σMIR and saturates, thus validating the MIR limit. However, in bad metals the relationship σ−1T continues unabated through the MIR limit. According to the Drude model these metals have lifetimes so short that the quasiparticles should be unstable (that is, decohere), producing an insulating state, and yet the transport properties remain metallic. Bad-metal conductivity is often found in strongly correlated materials such as the high-T superconductors and the Mott insulator–metal transition (IMT) system VO2. The phenomenon of bad-metal conductivity is a central problem in condensed matter physics1,2,3.

Here we study bad-metal conductivity and the insulator–metal transition in the nickelates. The nickelate phase diagram features an antiferromagnetic insulator at low T and a paramagnetic metal (PM) at high T. For R= Sm and heavier there is also an intermediate paramagnetic insulator for TN<T <TIM (TN= Néel temperature, TIM= IMT temperature)4. TIM and the measure of correlation in the metal phase (for example, the suppression of the coherent spectral weight) can be tuned by cation substitution, epitaxial strain, or dimensionality control4,5,6,7. The nickelates have strong interactions between the charge, spin, orbital and lattice degrees of freedom, and this complexity has obscured the physics of the IMT. Several distinct scenarios have been proposed, including a Mott transition, charge and spin density wave formation, orbital order, and a polaronic insulator8,9,10,11,12,13,14. Experiments show that the metallic phase is consistent with strong electronic correlations and that the electron–phonon interaction is important, but have not presented a clear explanation of the IMT (refs 5, 8, 11 and14). Recently, both total-energy theory and a model Hamiltonian theory have largely reproduced the phase diagram15,16.

To directly address the apparent breakdown of the lifetime hypothesis in a bad metal we measure a series of samples with controlled levels of disorder. We grow SmNiO3−δ epitaxial thin films on LaAlO3 by high-pressure sputtering. By varying the sputtering pressure we can control the concentration of oxygen vacancies, producing films with different amounts of disorder17,18. Here we report results for five samples, here named SNO1–SNO5, with increasing amounts of disorder. SNO1 is the closest to stoichiometric and displays a sharp IMT. SNO5 is the farthest from stoichiometric and displays a broadened IMT.

In Fig. 1a we present σDC(T) for a conventional metal (Cu), a saturating metal (Ti0.89Al0.11) and RNiO3. Cu is a good conductor at all T: both σDC and the fractional slope −dln(σ)/dT (Fig. 1b) are large19. Ti0.89Al0.11 is a saturating metal: σDC saturates slightly above σMIR at high T, and the slope −dln(σ)/dT is 30 times smaller than for Cu1. We also plot our measurements on SNO3 at high T along with previously published data for RNiO3, all in the PM phase. The published data include R= La, Pr, Nd, Sm and Gd, as well as doped compounds. For all RNiO3, σDC is below σMIR at high T. Importantly, the fractional slope −dln(σ)/dT is large for all RNiO3. For SNO3, the fractional slope at 800 K is almost identical to that of Cu, although σDC is orders of magnitude lower. This means that σDC does not saturate even as it falls below the MIR limit. Fig. 1 establishes RNiO3 as a class of bad metals.

Figure 1: Bad-metal conductivity in the high-T PM phase of the nickelates.
figure 1

We show our data for SNO3 along with published data for RNiO3—see Supplementary Information for a key to the symbols. We also show data for a conventional metal (Cu), and a saturating metal (Ti0.89Al0.11). a, σDC at high T. There is a clear separation of scales between good metals such as Cu, saturating metals such as Ti0.89Al0.11, and bad metals such as RNiO3. For RNiO3 we estimate σMIR 2,500 Ω−1 cm−1 (see Supplementary Information). b, Fractional derivative −dln(σDC)/ dT corresponding to the data in a. Both the bad metals RNiO3 and the conventional metal Cu have a relatively large derivative at high T. In contrast, the saturation of σDC for Ti0.89Al0.11 results in a fractional derivative that is suppressed by a factor of 10–100 compared to Cu and RNiO3. References for published data on RNiO3 are listed in the Supplementary Information, and the points represent the highest reported T.

We probe the complex infrared conductivity (σ(E)=σ (E)+i×σ ′′(E)) in the range E = 25–1,550 meV for T between 295 and 415 K. We measured and analysed 69 complete spectra with varying T and disorder; see Supplementary Information. We show in Fig. 2 the T-dependence of σ (E) for SNO1. The IMT is accompanied by a large increase in spectral weight below 400 meV, only some of which is transferred from the mid-infrared peak near 550 meV. The Ni–O bending and stretching modes (near 35 and 75 meV, respectively) are narrow phonons for T <TIM but merge into the coherent quasiparticle response in the PM phase, becoming strongly damped for T >TIM. We identify these phonons by referring to a general analysis of infrared-active phonons in distorted (Pnma) perovskites20. The Ni–O stretching mode is associated with a breathing distortion of the NiO6 octahedra, and the bending mode affects the Ni–O–Ni bond angle12. The quasiparticles in the PM phase are polarons11,12; our data identify the phonon modes that strongly couple with the electronic degrees of freedom. The emergence of a strongly damped phonon response forT >TIM also explains the far-infrared minimum in σ (ω), which is present in our data and is commonly observed in manganites and bad metals2.

Figure 2: σ (E) measured across the insulator-metal transition for SNO1.
figure 2

TIM as measured by the change in slope of σDC(T) is marked on the colour bar. (Inset) T-dependence of the mid-infrared peak position for all five samples SNO1–5 (solid circle, plus, asterisk, cross, open square, respectively).

The mid-infrared peak position (Fig. 2, inset) has the T-dependence expected for an order parameter and has been identified with Eg (ref. 21). However, instead of disappearing in the PM phase it persists as a shoulder on the broad quasiparticle response. We identify this shoulder as a Holstein side band, which is associated with the breakup of a polaron into its constituent phonon and electron degrees of freedom22. The connection of the Eg-like feature for T <TIM to the Holstein side band for T >TIM suggests that the IMT is driven by the freezing of the same lattice distortion that is responsible for polaron formation in the PM phase. This is consistent with experiments showing a reduction in symmetry from orthorhombic for T >TIM to monoclinic for T <TIM and with theory showing that the insulating phase is controlled by a period-doubling lattice distortion15,16,23,24.

The effects of varying T and disorder are captured by the spectral weight:

f(E) counts the quasiparticles that contribute to conductivity at frequencies ω <E/. f(EW), where W is the conduction band width, measures the effect of strong coupling on the electronic structure. The plasma frequency (Ωp) is:

where ɛ0 is the dielectric permittivity of free space. Strong coupling is taken to be any interaction that significantly alters n or m in the PM phase.

We plot in Fig. 3a the T-dependence of f(E) for SNO1. The IMT is accompanied by a steep rise in f(E) in the range E < 400 meV, signalling the collapse of Eg and the filling of the conduction band. Much of the spectral weight transfer comes from high-energy states beyond 1.5 eV, as observed for other RNiO3 (refs 5, 8). ForT >TIM, f(E) is nearly indistinguishable from that measured on 2.9% tensile strained LaNiO3-on-DyScO3 at room temperature (ref. 5). This suggests a common mechanism for spectral weight suppression for epitaxial strain and rare-earth substitution in RNiO3.

Figure 3: Spectral weight and extended Drude analysis.
figure 3

a, Solid lines: f(E) for SNO1 across the IMT; the colour scale is the same as in Fig. 2. Broken lines: f(E) for epitaxially strained LaNiO3 (ref. 5; red =−1.1% strain on LaAlO3, blue =+0.6% strain on (LaAlO3)0.3(Sr2AlTaO6)0.7; green =+2.9% strain on DyScO3). b,c, Extended Drude analysis for SNO1–SNO5. Symbols are the same as in Fig. 2. Data is significant only in the PM phase (T >TIM).TIM= 394, 386, 380, 387 and 374 K for SNO1–5, respectively. This range of TIM values is shown by the dashed vertical lines. In b, τ−1(ω→0) is nearly constant for all samples, irrespective of T or disorder. In c, m(ω→0)/mb is independent of T for T >TIM for all samples (note the variation in TIM).

We plot in Fig. 3b–c results from an extended Drude analysis of all samples (Supplementary Information). For an IMT driven by electron–electron correlation m diverges as TTIM+, as observed in Mott insulators such as VO2 (ref. 3). For SmNiO3 we observe no such mass enhancement approaching TIM. In a conventional metal the effect of adding disorder is to decrease τ, thereby lowering σDC. For SmNiO3 we see that τ(ω→0) is nearly constant for all samples, irrespective of the amount of disorder.

According to the standard Drude model if both m and τ are constant then a changing σDC must result from a changing n. In Fig. 4a we overlay σDC(T) for SNO1–5 with Ωp2 (Supplementary Information). Ωp2 responds to changes in T and disorder in nearly the same way as σDC(T). The correspondence in the PM phase is remarkable: Ωp2 decreases in response to increased T or disorder, just as τ does in a conventional metal (Fig. 4b). Fig. 4 suggests an explanation for bad-metal conductivity in the nickelates: small changes in T or disorder produce continuous shifts of spectral weight over a large energy scale. This stands in stark contrast to normal metals in which changes in T or disorder affect τ. The defining characteristic of bad metals is the presence of a strong interaction that can continuously shift spectral weight between the conduction band and high-energy states in response to small changes in T and disorder, yielding metallic conductivity below σMIR while preserving the Drude model and without violating the uncertainty principle2.

Figure 4: Spectral weight shifts in response to T, disorder and the insulator-metal transition.
figure 4

a, σDC (left axis, lines) and Ωp2 (right axis, squares) for SNO1–5, (red, green, purple, teal, blue, respectively) as a function of T across TIM. In the PM phase Ωp2 decreases in response to increased T or disorder, just as τ does in a conventional metal. b, Schematic showing the different responses of a bad metal and a conventional metal to increases in T or disorder. c, Schematic of the bending and breathing distortions that shift spectral weight by altering the Ni–O covalence. Blue is O and green is Ni. We draw select Ni eg and O 2p orbitals to illustrate how the distortions affect orbital overlap. d, Schematic of the electronic structure; emphasis is on Ni–O bonding and how the Ni eg and O 2p orbitals bond in the covalent and ionic limits. Outer columns correspond to the electronic structure in the nominal t2g6eg1 configuration. The inner-left column shows the limit of covalent Ni–O bonding, with a high energy separating the pdσ and pdσ states15,27. The inner-right column shows the limits of ionic Ni–O bonding, with the eg2 and L states separated by the (negative) charge transfer energy.

This high-energy scale is provided by the interaction between Ni–O bond covalence and local lattice distortions. The ground state electronic configuration of RNiO3 is nominally t2g6eg1 (Ni3+). However, this ionic representation is inaccurate owing to the strong covalent sigma bonds between Ni eg and O 2p orbitals, and the RNiO3 family are best described as negative charge-transfer valence bond insulators25,26. Theory shows that the low-energy density of states (DOS) near the Fermi energy (EF) is antibonding in character, the bonding DOS is centred at 6–8 eV below EF, and this separation is the dominant high-energy feature in the electronic structure15,27. Importantly, Ni–O covalence is controlled by lattice distortions (Fig. 4c–d)25. Overlap between the Ni eg and O 2p orbitals depends on the Ni–O bond lengths and the Ni–O–Ni bond angles. The bond lengths are affected by the breathing mode, and the bond angles are affected by the bending mode. The high-energy electronic structure of RNiO3 is therefore controlled by the local geometry of the NiO6 octahedra.

Both the bending and the breathing modes condense at TIM, resulting in a discrete change in the Ni–O–Ni bond angle and a disproportionation into expanded (Ni1 sites) and contracted (Ni2 sites) octahedra14,15,16,23,24. This affects the Ni–O covalence and transfers spectral weight over large energies, as observed experimentally8. In the insulating phases the Ni1 sites are closer to the ionic limit t2g6eg2L (L denotes a ‘ligand hole’ on an oxygen orbital), with spin quantum number S = 1. The Ni2 sites are closer to the covalent limit, with electrons in bonding orbitals forming S = 0 singlets13,14,15. Eg is approximately ten times smaller than the large-energy scales in the system, and results from period doubling that splits the low-energy DOS (refs 9, 13, 15). The low-energy DOS is predominantly L-like; transport through the eg2 manifold is suppressed by the Hubbard intra-orbital electron–electron Coulomb interaction.

In the PM phase, spectral weight is continuously redistributed in response to changes in T or disorder by dynamic electron–phonon interactions, as suggested by the strongly damped bending and breathing modes and by the observation of dynamic Ni1–Ni2 fluctuations above TIM (ref. 24). It is a challenge to theory to fully describe the electron–phonon interaction in the PM phase. A full description will probably include a pseudogap in the metallic DOS. It would be interesting to study the effects of controllably introducing different types of disorder on the spectral weight and the lattice distortion in the PM phase of the nickelates.

The shifting of spectral weight in the PM phase is also seen in Mott–Hubbard materials2. We conclude that a strong interaction that can shift spectral weight over a large energy scale is necessary for bad-metal conductivity, but that the identity of this interaction may vary. Notably, our work describes bad-metal conductivity in a material system in which the Hubbard interaction is of secondary importance. Both total energy and model Hamiltonian theory agree that the Hubbard interaction does not control the phase diagram, it does not significantly affect the electronic structure, and it is not the driving force for the IMT in the nickelates15,16,27. Our results confirm the primary importance of the electron–phonon interaction and rule out a conventional Mott–Hubbard IMT. This is consistent with the Zaanen–Sawatzky–Allen classification of transition metal oxides, wherein for late transition metals (including Ni) the Hubbard energy is so large that it only indirectly controls the phase diagram28.